We generalize the application of the functional renormalization group (fRG) method for the fermionic flow into the symmetry-broken phase to finite temperatures. We apply the scheme to the case of a broken discrete symmetry: the charge-density wave (CDW) mean-field model at half filling. We show how an arbitrarily small initial CDW order parameter starts to grow at the CDW instability and how it flows to the correct final value, suppressing the divergence of the effective interaction in the fRG flow. The effective interaction peaks at the instability and saturates at low energy scales or temperatures. The relation to the mean-field treatment, differences compared to the flow for a broken continuous symmetry, and the prospects of the new method are discussed.